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278                     ANALYTICAL MECHANICS
Making these substitutions
p = l mg cos 6. ' Q = j nig sin 6.
4 tan < = -fe tan 0.
DISCUSSION.  The reaction and its direction are independent of length of the rod. When 0 = 0, Q = 0 and R = P = S w0r. In cr words at the instant when the rod passes the lowest point the force on
axis is f times as large as the force when rod hangs at rest.    When 0 =
P = 0 and R = Q = Img. If the rod is held in a horizontal positior supporting the free end the reaction of the axis is J mg. But as soo; the support is removed from the free end the reaction on the axis is chai from  mg to i mg.
PROBLEMS.
1.   A uniform rod which is free to rotate about a horizontal axis ; from the position of unstable equilibrium.   Find the reaction of the i
2.   In the preceding problem find the position where the horizo component of the reaction is a maximum.
3.   A uniform rod which is free to rotate about a horizontal axis i from a horizontal position.    Show that the horizontal component of reaction is greatest when the rod makes 45 with the vertical.
4.   A cube rotates about a horizontal axis which coincides with on its edges.   If at the highest position it barely completes the revolut
i,      xu x n     3  5cos0TTr      i *     sin 0 ir     ,       1ir .   , .        , , show that P = - - - W and Q = - . - \v , where \V is the weigh 2>                                 HC
the cube.
6. A cube which is free to rotate about a horizontal axis through of its edges starts to fall when its center is at the same level as the axi rotation. Find the reaction of the axis.
6.   Show that if the body of 214 is a particle connected to the ; with a massless rod the reaction perpendicular to the rod vanishes.
7.   Consider the reactions of the axis when the latter pauses thro the center of mass of the rigid body.
8.   A circular plate is free to rotate about a horizontal axis which fo one of the elements of its cylindrical surface.   The plate is let fall fi the position when its center of mass is vertically above the axis.   .
termine the reaction of the axis at 6 = ^ and at 6 = 0.